Preheat the oven to 350 degrees. Line a baking sheet with parchment.
Put the butter, sugar and cinnamon into a bowl and mix well. Stand a doughnut up on its side and slice it diagonally across the middle. Make an incision along the ring of the doughnut and fill with cinnamon butter. Repeat with the rest. Arrange the doughnut halves in a ring on the baking sheet and bake for 15 minutes.
Put the powdered sugar and milk into a small bowl and mix well. Drizzle over the hot doughnut ring and serve warm.